From: Andrea Asperti <andrea.asperti@unibo.it>
Date: Mon, 14 Oct 2013 18:08:22 +0000 (+0000)
Subject: almost there
X-Git-Tag: make_still_working~1080
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=2ee1d9b1e0589aea7c5ca39a4b2a208b47f6463e;p=helm.git

almost there
---

diff --git a/matita/matita/lib/turing/multi_to_mono/multi_to_mono.ma b/matita/matita/lib/turing/multi_to_mono/multi_to_mono.ma
index c0c102694..0e9bce53e 100644
--- a/matita/matita/lib/turing/multi_to_mono/multi_to_mono.ma
+++ b/matita/matita/lib/turing/multi_to_mono/multi_to_mono.ma
@@ -7,13 +7,71 @@ include "turing/multi_to_mono/shift_trace_machines.ma".
 (* come back to the head position.                                            *)
 (******************************************************************************)
 
+(* we say that a tape is regular if for any trace after the first blank we
+  only have other blanks *)
+  
+definition all_blanks_in ≝ λsig,l.
+  ∀x. mem ? x l → x = blank sig.  
+ 
+definition regular_i  ≝ λsig,n.λl:list (multi_sig sig n).λi.
+  all_blanks_in ? (after_blank ? (trace sig n i l)).
+
+definition regular_trace ≝ λsig,n,a.λls,rs:list (multi_sig sig n).λi.
+  Or (And (regular_i sig n (a::ls) i) (regular_i sig n rs i))
+     (And (regular_i sig n ls i) (regular_i sig n (a::rs) i)).
+         
+axiom regular_tail: ∀sig,n,l,i.
+  regular_i sig n l i → regular_i sig n (tail ? l) i.
+  
+axiom regular_extend: ∀sig,n,l,i.
+   regular_i sig n l i → regular_i sig n (l@[all_blank sig n]) i.
+
+axiom all_blank_after_blank: ∀sig,n,l1,b,l2,i. 
+  nth i ? (vec … b) (blank ?) = blank ? → 
+  regular_i sig n (l1@b::l2) i → all_blanks_in ? (trace sig n i l2).
+   
+lemma regular_trace_extl:  ∀sig,n,a,ls,rs,i.
+  regular_trace sig n a ls rs i → 
+    regular_trace sig n a (ls@[all_blank sig n]) rs i.
+#sig #n #a #ls #rs #i *
+  [* #H1 #H2 % % // @(regular_extend … H1)
+  |* #H1 #H2 %2 % // @(regular_extend … H1)
+  ]
+qed.
+
+lemma regular_cons_hd_rs: ∀sig,n.∀a:multi_sig sig n.∀ls,rs1,rs2,i.
+  regular_trace sig n a ls (rs1@rs2) i → 
+    regular_trace sig n a ls (rs1@((hd ? rs2 (all_blank …))::(tail ? rs2))) i.
+#sig #n #a #ls #rs1 #rs2 #i cases rs2 [2: #b #tl #H @H] 
+*[* #H1 >append_nil #H2 %1 % 
+   [@H1 | whd in match (hd ???); @(regular_extend … rs1) //]
+ |* #H1 >append_nil #H2 %2 % 
+   [@H1 | whd in match (hd ???); @(regular_extend … (a::rs1)) //]
+ ]
+qed.
+
+lemma eq_trace_to_regular : ∀sig,n.∀a1,a2:multi_sig sig n.∀ls1,ls2,rs1,rs2,i.
+   nth i ? (vec … a1) (blank ?) = nth i ? (vec … a2) (blank ?) →
+   trace sig n i ls1 = trace sig n i ls2 → 
+   trace sig n i rs1 = trace sig n i rs2 →
+   regular_trace sig n a1 ls1 rs1 i → 
+     regular_trace sig n a2 ls2 rs2 i.
+#sig #n #a1 #a2 #ls1 #ls2 #rs1 #rs2 #i #H1 #H2 #H3 #H4
+whd in match (regular_trace ??????); whd in match (regular_i ????);
+whd in match (regular_i ?? rs2 ?); whd in match (regular_i ?? ls2 ?);
+whd in match (regular_i ?? (a2::rs2) ?); whd in match (trace ????);
+<trace_def whd in match (trace ??? (a2::rs2)); <trace_def 
+<H1 <H2 <H3 @H4
+qed.
+
 (******************************* move_to_blank_L ******************************)
 (* we compose machines together to reduce the number of output cases, and 
    improve semantics *)
    
 definition move_to_blank_L ≝ λsig,n,i.
   (move_until ? L (no_blank sig n i)) · extend ? (all_blank sig n).
-  
+
+(*
 definition R_move_to_blank_L ≝ λsig,n,i,t1,t2.
 (current ? t1 = None ? → 
   t2 = midtape (multi_sig sig n) (left ? t1) (all_blank …) (right ? t1)) ∧
@@ -22,18 +80,25 @@ definition R_move_to_blank_L ≝ λsig,n,i,t1,t2.
   (∃b,ls1,ls2.
     (no_blank sig n i b = false) ∧ 
     (∀j.j ≤n → to_blank_i ?? j (ls1@b::ls2) = to_blank_i ?? j ls) ∧
-    t2 = midtape ? ls2 b ((reverse ? (a::ls1))@rs)).
+    t2 = midtape ? ls2 b ((reverse ? (a::ls1))@rs)). 
+*)
 
-definition R_move_to_blank_L_new ≝ λsig,n,i,t1,t2.
+definition R_move_to_blank_L ≝ λsig,n,i,t1,t2.
 (current ? t1 = None ? → 
   t2 = midtape (multi_sig sig n) (left ? t1) (all_blank …) (right ? t1)) ∧
-∀ls,a,rs.t1 = midtape ? ls a rs → 
+∀ls,a,rs.
+  t1 = midtape (multi_sig sig n) ls a rs → 
+  regular_i sig n (a::ls) i →
+  (∀j. j ≠ i → regular_trace … a ls rs j) →
   (∃b,ls1,ls2.
+    (regular_i sig n (ls1@b::ls2) i) ∧ 
+    (∀j. j ≠ i → regular_trace … 
+      (hd ? (ls1@b::ls2) (all_blank …)) (tail ? (ls1@b::ls2)) rs j) ∧ 
     (no_blank sig n i b = false) ∧ 
-    (hd ? (ls1@[b]) (all_blank …) = a) ∧ (* not implied by the next fact *)
+    (hd (multi_sig sig n) (ls1@[b]) (all_blank …) = a) ∧ (* not implied by the next fact *)
     (∀j.j ≤n → to_blank_i ?? j (ls1@b::ls2) = to_blank_i ?? j (a::ls)) ∧
     t2 = midtape ? ls2 b ((reverse ? ls1)@rs)).
-    
+
 theorem sem_move_to_blank_L: ∀sig,n,i. 
   move_to_blank_L sig n i  ⊨ R_move_to_blank_L sig n i.
 #sig #n #i 
@@ -41,47 +106,35 @@ theorem sem_move_to_blank_L: ∀sig,n,i.
   (ssem_move_until_L ? (no_blank sig n i)) (sem_extend ? (all_blank sig n)))
 #tin #tout * #t1 * * #Ht1a #Ht1b * #Ht2a #Ht2b %
   [#Hcur >(Ht1a Hcur) in Ht2a; /2 by /
-  |#ls #a #rs #Htin -Ht1a cases (Ht1b … Htin)
-    [* #Hnb #Ht1 -Ht1b -Ht2a >Ht1 in Ht2b; >Htin #H %1 
-     % [//|@H normalize % #H1 destruct (H1)] 
+  |#ls #a #rs #Htin #Hreg #Hreg2 -Ht1a cases (Ht1b … Htin)
+    [* #Hnb #Ht1 -Ht1b -Ht2a >Ht1 in Ht2b; >Htin #H 
+     %{a} %{[ ]} %{ls} 
+     %[%[%[%[%[@Hreg|@Hreg2]|@Hnb]|//]|//]|@H normalize % #H1 destruct (H1)]
     |* 
     [(* we find the blank *)
-     * #ls1 * #b * #ls2 * * * #H1 #H2 #H3 #H4 %2
-     %{b} %{ls1} %{ls2} 
-     %[%[@H2|>H1 //] 
-      |-Ht1b -Ht2a >H4 in Ht2b; #H5 @H5
-       % normalize #H6 destruct (H6)
+     * #ls1 * #b * #ls2 * * * #H1 #H2 #H3 #Ht1
+     >Ht1 in Ht2b; #Hout -Ht1b
+     %{b} %{(a::ls1)} %{ls2} 
+     %[%[%[%[%[>H1 in Hreg; #H @H 
+              |#j #jneqi whd in match (hd ???); whd in match (tail ??);
+               <H1 @(Hreg2 j jneqi)]|@H2] |//]|>H1 //]
+      |@Hout normalize % normalize #H destruct (H)
       ]
-    |* #b * #lss * * #H1 #H2 #H3 %2
-     %{(all_blank …)} %{ls} %{[ ]} 
-     %[%[whd in match (no_blank ????); >blank_all_blank // @daemon
-        |@daemon (* make a lemma *)] 
-      |-Ht1b -Ht2b >H3 in Ht2a; 
-       whd in match (left ??); whd in match (right ??); #H4
-       >H2 >reverse_append >reverse_single @H4 normalize // 
+    |* #b * #lss * * #H1 #H2 #Ht1 -Ht1b >Ht1 in Ht2a; 
+     whd in match (left ??); whd in match (right ??); #Hout
+     %{(all_blank …)} %{(lss@[b])} %{[]}
+     %[%[%[%[%[<H2 @regular_extend // 
+              |<H2 #j #jneqi whd in match (hd ???); whd in match (tail ??); 
+               @regular_trace_extl @Hreg2 //]
+            |whd in match (no_blank ????); >blank_all_blank //]
+          |<H2 //]
+        |#j #lejn <H2 @sym_eq @to_blank_i_ext]
+      |>reverse_append >reverse_single @Hout normalize // 
       ]
     ]
   ]
 qed.
 
-theorem sem_move_to_blank_L_new: ∀sig,n,i. 
-  move_to_blank_L sig n i  ⊨ R_move_to_blank_L_new sig n i.
-#sig #n #i 
-@(Realize_to_Realize … (sem_move_to_blank_L sig n i))
-#t1 #t2 * #H1 #H2 % [@H1] 
-#ls #a #rs #Ht1 lapply (H2 ls a rs Ht1) -H2 *
-  [* #Ha #Ht2 >Ht2 %{a} %{[ ]} %{ls} 
-   %[%[%[@Ha| //]| normalize //] | @Ht1]
-  |* #b * #ls1 * #ls2 * * * #Hblank #Ht2
-   %{b} %{(a::ls1)} %{ls2} 
-   %[2:@Ht2|%[%[//|//]|
-   #j #lejn whd in match (to_blank_i ????); 
-   whd in match (to_blank_i ???(a::ls));
-   lapply (Hblank j lejn) whd in match (to_blank_i ????);
-   whd in match (to_blank_i ???(a::ls)); #H >H %
-  ]
-qed.
-
 (******************************************************************************)
    
 definition shift_i_L ≝ λsig,n,i.
@@ -90,10 +143,6 @@ definition shift_i_L ≝ λsig,n,i.
    extend ? (all_blank sig n).
    
 definition R_shift_i_L ≝ λsig,n,i,t1,t2.
-  (* ∀b:multi_sig sig n.∀ls.
-    (t1 = midtape ? ls b [ ] → 
-     t2 = midtape (multi_sig sig n) 
-      ((shift_i sig n i b (all_blank sig n))::ls) (all_blank sig n) [ ]) ∧ *)
     (∀a,ls,rs. 
       t1 = midtape ? ls a rs  → 
       ((∃rs1,b,rs2,a1,rss.
@@ -126,10 +175,6 @@ theorem sem_shift_i_L: ∀sig,n,i. shift_i_L sig n i  ⊨ R_shift_i_L sig n i.
    (sem_seq ????? (ssem_mti sig n i) 
     (sem_extend ? (all_blank sig n))))
 #tin #tout * #t1 * * #Ht1a #Ht1b * #t2 * * #Ht2a #Ht2b * #Htout1 #Htout2
-(* #b #ls %
-  [#Htin lapply (Ht1a … Htin) -Ht1a -Ht1b #Ht1 
-   lapply (Ht2a … Ht1) -Ht2a -Ht2b #Ht2 >Ht2 in Htout1; 
-   >Ht1 whd in match (left ??); whd in match (right ??); #Htout @Htout // *)
 #a #ls #rs cases rs
   [#Htin %2 %{(shift_i sig n i a (all_blank sig n))} %{[ ]} 
    %[%[#x @False_ind | @daemon]
@@ -182,7 +227,12 @@ theorem sem_shift_i_L_new: ∀sig,n,i.
 qed.
    
 
-(******************************************************************************)
+(*******************************************************************************
+The following machine implements a full move of for a trace: we reach the left 
+border, shift the i-th trace and come back to the head position. *)  
+
+(* this exclude the possibility that traces do not overlap: the head must 
+remain inside all traces *)
 
 definition mtiL ≝ λsig,n,i.
    move_to_blank_L sig n i · 
@@ -192,11 +242,16 @@ definition mtiL ≝ λsig,n,i.
 definition Rmtil ≝ λsig,n,i,t1,t2.
   ∀ls,a,rs. 
    t1 = midtape (multi_sig sig n) ls a rs → 
-   nth n ? (vec … a) (blank ?) = head ? →   
+   nth n ? (vec … a) (blank ?) = head ? →  
+   (∀i.regular_trace sig n a ls rs i) → 
+   (* next: we cannot be on rightof on trace i *)
+   (nth i ? (vec … a) (blank ?) = (blank ?) 
+     → nth i ? (vec … (hd ? rs (all_blank …))) (blank ?) ≠ (blank ?)) →
    no_head_in … ls →   
    no_head_in … rs  → 
    (∃ls1,a1,rs1.
      t2 = midtape (multi_sig …) ls1 a1 rs1 ∧
+     (∀i.regular_trace … a1 ls1 rs1 i) ∧
      (∀j. j ≤ n → j ≠ i → to_blank_i ? n j (a1::ls1) = to_blank_i ? n j (a::ls)) ∧
      (∀j. j ≤ n → j ≠ i → to_blank_i ? n j rs1 = to_blank_i ? n j rs) ∧
      (to_blank_i ? n i ls1 = to_blank_i ? n i (a::ls)) ∧
@@ -205,47 +260,53 @@ definition Rmtil ≝ λsig,n,i,t1,t2.
 theorem sem_Rmtil: ∀sig,n,i. i < n → mtiL sig n i  ⊨ Rmtil sig n i.
 #sig #n #i #lt_in
 @(sem_seq_app ?????? 
-  (sem_move_to_blank_L_new … )
+  (sem_move_to_blank_L … )
    (sem_seq ????? (sem_shift_i_L_new …)
     (ssem_move_until_L ? (no_head sig n))))
 #tin #tout * #t1 * * #_ #Ht1 * #t2 * #Ht2 * #_ #Htout
 (* we start looking into Rmitl *)
 #ls #a #rs #Htin (* tin is a midtape *)
-#Hhead #Hnohead_ls #Hnohead_rs  
-lapply (Ht1 … Htin) -Ht1 -Htin
-* #b * #ls1 * #ls2 * * * #Hno_blankb #Hhead #Hls1 #Ht1
+#Hhead #Hreg #no_rightof #Hnohead_ls #Hnohead_rs 
+cut (regular_i sig n (a::ls) i)
+  [cases (Hreg i) * // 
+   cases (true_or_false (nth i ? (vec … a) (blank ?) == (blank ?))) #Htest
+    [#_ @daemon (* absurd, since hd rs non e' blank *)
+    |#H #_ @daemon]] #Hreg1
+lapply (Ht1 … Htin Hreg1 ?) [#j #_ @Hreg] -Ht1 -Htin
+* #b * #ls1 * #ls2 * * * * * #reg_ls1_i #reg_ls1_j #Hno_blankb #Hhead #Hls1 #Ht1
 lapply (Ht2 … Ht1) -Ht2 -Ht1 
 * #rs1 * #b0 * #rs2 * #rss * * * * * #Hb0 #Hb0blank #Hrs1 #Hrs1b #Hrss #Ht2
 (* we need to recover the position of the head of the emulated machine
    that is the head of ls1. This is somewhere inside rs1 *) 
 cut (∃rs11. rs1 = (reverse ? ls1)@rs11)
-     [cut (ls1 = [ ] ∨ ∃aa,tlls1. ls1 = aa::tlls1)
-       [cases ls1 [%1 // | #aa #tlls1 %2 %{aa} %{tlls1} //]] *
-       [#H1ls1 %{rs1} >H1ls1 //
-       |* #aa * #tlls1 #H1ls1 >H1ls1 in Hrs1; 
-        cut (aa = a) [>H1ls1 in Hls1; #H @(to_blank_hd … H)] #eqaa >eqaa  
-        #Hrs1_aux cases (compare_append … (sym_eq … Hrs1_aux)) #l *
-         [* #H1 #H2 %{l} @H1
-         |(* this is absurd : if l is empty, the case is as before.
+  [cut (ls1 = [ ] ∨ ∃aa,tlls1. ls1 = aa::tlls1)
+    [cases ls1 [%1 // | #aa #tlls1 %2 %{aa} %{tlls1} //]] *
+      [#H1ls1 %{rs1} >H1ls1 //
+      |* #aa * #tlls1 #H1ls1 >H1ls1 in Hrs1; 
+       cut (aa = a) [>H1ls1 in Hls1; #H @(to_blank_hd … H)] #eqaa >eqaa  
+       #Hrs1_aux cases (compare_append … (sym_eq … Hrs1_aux)) #l *
+        [* #H1 #H2 %{l} @H1
+        |(* this is absurd : if l is empty, the case is as before.
            if l is not empty then it must start with a blank, since it is the
-           frist character in rs2. But in this case we would have a blank 
-           inside ls1=attls1 that is absurd *)
+           first character in rs2. But in this case we would have a blank 
+           inside ls1=a::tls1 that is absurd *)
           @daemon
-         ]]]   
+        ]]]   
    * #rs11 #H1
+cut (rs = rs11@rs2)
+  [@(injective_append_l … (reverse … ls1)) >Hrs1 <associative_append <H1 //] #H2
 lapply (Htout … Ht2) -Htout -Ht2 *
-  [(* the current character on trace i hold the head-mark.
+  [(* the current character on trace i holds the head-mark.
       The case is absurd, since b0 is the head of rs2, that is a sublist of rs, 
       and the head-mark is not in rs *)
-   @daemon
-   (* something is missing 
-   * #H @False_ind @(absurd ? H) @eqnot_to_noteq whd in ⊢ (???%);
-   cut (rs2 = [ ] ∨ ∃c,rs21. rs2 = c::rs21)
-    [cases rs2 [ %1 // | #c #rs22 %2 %{c} %{rs22} //]] 
-   * (* by cases on rs2 *)
-    [#Hrs2 >Hb0 >Hrs2 normalize >all_blank_n //
-    |* #c * #rs22 #Hrs2 destruct (Hrs2) @no_head_true @Hnohead_rs
-     > *)
+   * #H3 @False_ind @(absurd (nth n ? (vec … b0) (blank sig) = head ?)) 
+     [@(\P ?) @injective_notb @H3 ]
+   @Hnohead_rs >H2 >trace_append @mem_append_l2 
+   lapply Hb0 cases rs2 
+    [whd in match (hd ???); #H >H in H3; whd in match (no_head ???); 
+     >all_blank_n normalize -H #H destruct (H); @False_ind
+    |#c #r #H4 %1 >H4 //
+    ]
   |* 
   [(* we reach the head position *)
    (* cut (trace sig n j (a1::ls20)=trace sig n j (ls1@b::ls2)) *)
@@ -275,24 +336,40 @@ lapply (Htout … Ht2) -Htout -Ht2 *
       | *) ] #H2
   cut ((trace sig n i (b0::reverse ? rs11) = trace sig n i (ls10@[a1])) ∧ 
       (trace sig n i (ls1@ls2) = trace sig n i ls20))
-    [>H1 in Htracei; >reverse_append >reverse_single >reverse_append 
+   [>H1 in Htracei; >reverse_append >reverse_single >reverse_append 
      >reverse_reverse >associative_append >associative_append
      @daemon
-    ] #H3
-   %{ls20} %{a1} %{(reverse ? (b0::ls10)@tail (multi_sig sig n) rs2)}
-   %[%[%[%[@Htout
-    |#j #lejn #jneqi <(Hls1 … lejn) 
-     >to_blank_i_def >to_blank_i_def @eq_f @sym_eq @(proj2 … (H2 j jneqi))]
-    |#j #lejn #jneqi >reverse_cons >associative_append >Hb0
-     <to_blank_hd_cons >to_blank_i_def >to_blank_i_def @eq_f
-     @(injective_append_l … (trace sig n j (reverse ? ls1))) (* >trace_def >trace_def *)
+    ] #H3    
+  cut (∀j. j ≠ i → 
+    trace sig n j (reverse (multi_sig sig n) ls10@rs2) = trace sig n j rs)
+    [#j #jneqi @(injective_append_l … (trace sig n j (reverse ? ls1)))
      >map_append >map_append >Hrs1 >H1 >associative_append 
      <map_append <map_append in ⊢ (???%); @eq_f 
      <map_append <map_append @eq_f2 // @sym_eq 
      <(reverse_reverse … rs11) <reverse_map <reverse_map in ⊢ (???%);
-     @eq_f @(proj1 … (H2 j jneqi))]
-    |<(Hls1 i) [2:@lt_to_le //] (* manca un invariante: dopo un blank
-     posso avere solo blank *) @daemon ]
+     @eq_f @(proj1 … (H2 j jneqi))] #Hrs_j
+   %{ls20} %{a1} %{(reverse ? (b0::ls10)@tail (multi_sig sig n) rs2)}
+   %[%[%[%[%[@Htout
+    |#j cases (decidable_eq_nat j i)
+      [#eqji >eqji (* by cases wether a1 is blank *)
+       @daemon
+      |#jneqi lapply (reg_ls1_j … jneqi) #H4 
+       >reverse_cons >associative_append >Hb0 @regular_cons_hd_rs
+       @(eq_trace_to_regular … H4)
+        [<hd_trace >(proj2 … (H2 … jneqi)) >hd_trace %
+        |<tail_trace >(proj2 … (H2 … jneqi)) >tail_trace %
+        |@sym_eq @Hrs_j //
+        ]
+      ]]
+    |#j #lejn #jneqi <(Hls1 … lejn) 
+     >to_blank_i_def >to_blank_i_def @eq_f @sym_eq @(proj2 … (H2 j jneqi))]
+    |#j #lejn #jneqi >reverse_cons >associative_append >Hb0
+     <to_blank_hd_cons >to_blank_i_def >to_blank_i_def @eq_f @Hrs_j //]
+    |<(Hls1 i) [2:@lt_to_le //] 
+     lapply (all_blank_after_blank … reg_ls1_i) 
+       [@(\P ?) @daemon] #allb_ls2
+     whd in match (to_blank_i ????); <(proj2 … H3)
+     @daemon ]
     |>reverse_cons >associative_append  
      cut (to_blank_i sig n i rs = to_blank_i sig n i (rs11@[b0])) [@daemon] 
      #Hcut >Hcut >(to_blank_i_chop  … b0 (a1::reverse …ls10)) [2: @Hb0blank]
diff --git a/matita/matita/lib/turing/multi_to_mono/trace_alphabet.ma b/matita/matita/lib/turing/multi_to_mono/trace_alphabet.ma
index 86db47400..bb1e54e63 100644
--- a/matita/matita/lib/turing/multi_to_mono/trace_alphabet.ma
+++ b/matita/matita/lib/turing/multi_to_mono/trace_alphabet.ma
@@ -57,6 +57,12 @@ definition trace ≝ λsig,n,i,l.
 lemma trace_def : ∀sig,n,i,l. 
   trace sig n i l = map ?? (λa. nth i ? (vec … n a) (blank sig)) l.
 // qed.
+
+lemma hd_trace:  ∀sig,n,i,l. 
+  hd ? (trace sig n i l) (blank ?) = 
+    nth i ? (hd ? l (all_blank … )) (blank ?).
+#sig #n #i #l elim l // normalize >blank_all_blank % 
+qed.
  
 lemma tail_trace:  ∀sig,n,i,l. 
   tail ? (trace sig n i l) = trace sig n i (tail ? l).
@@ -95,6 +101,7 @@ qed.
 definition no_head_in ≝ λsig,n,l. 
   ∀x. mem ? x (trace sig n n l) → x ≠ head ?.
 
+
 (* some lemmas over lists, to be moved *)
 lemma injective_append_l: ∀A,l. injective ?? (append A l).
 #A #l elim l 
@@ -212,6 +219,12 @@ let rec to_blank sig l on l ≝
   | cons a tl ⇒ 
       if a == blank sig then [ ] else a::(to_blank sig tl)]. 
       
+let rec after_blank sig l on l ≝
+  match l with
+  [ nil ⇒  [ ]
+  | cons a tl ⇒ 
+      if a == blank sig then (a::tl) else (after_blank sig tl)]. 
+      
 definition to_blank_i ≝ λsig,n,i,l. to_blank ? (trace sig n i l).
 
 lemma to_blank_i_def : ∀sig,n,i,l.